| from ._basic import _dispatch |
| from scipy._lib.uarray import Dispatchable |
| import numpy as np |
|
|
| __all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn'] |
|
|
|
|
| @_dispatch |
| def dctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, |
| workers=None, *, orthogonalize=None): |
| """ |
| Return multidimensional Discrete Cosine Transform along the specified axes. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DCT (see Notes). Default type is 2. |
| s : int or array_like of ints or None, optional |
| The shape of the result. If both `s` and `axes` (see below) are None, |
| `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is |
| ``numpy.take(x.shape, axes, axis=0)``. |
| If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``s[i] < x.shape[i]``, the ith dimension is truncated to length |
| ``s[i]``. |
| If any element of `s` is -1, the size of the corresponding dimension of |
| `x` is used. |
| axes : int or array_like of ints or None, optional |
| Axes over which the DCT is computed. If not given, the last ``len(s)`` |
| axes are used, or all axes if `s` is also not specified. |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized DCT variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| y : ndarray of real |
| The transformed input array. |
| |
| See Also |
| -------- |
| idctn : Inverse multidimensional DCT |
| |
| Notes |
| ----- |
| For full details of the DCT types and normalization modes, as well as |
| references, see `dct`. |
| |
| Examples |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fft import dctn, idctn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idctn(dctn(y))) |
| True |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
|
|
| @_dispatch |
| def idctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, |
| workers=None, orthogonalize=None): |
| """ |
| Return multidimensional Inverse Discrete Cosine Transform along the specified axes. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DCT (see Notes). Default type is 2. |
| s : int or array_like of ints or None, optional |
| The shape of the result. If both `s` and `axes` (see below) are |
| None, `s` is ``x.shape``; if `s` is None but `axes` is |
| not None, then `s` is ``numpy.take(x.shape, axes, axis=0)``. |
| If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``s[i] < x.shape[i]``, the ith dimension is truncated to length |
| ``s[i]``. |
| If any element of `s` is -1, the size of the corresponding dimension of |
| `x` is used. |
| axes : int or array_like of ints or None, optional |
| Axes over which the IDCT is computed. If not given, the last ``len(s)`` |
| axes are used, or all axes if `s` is also not specified. |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized IDCT variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| y : ndarray of real |
| The transformed input array. |
| |
| See Also |
| -------- |
| dctn : multidimensional DCT |
| |
| Notes |
| ----- |
| For full details of the IDCT types and normalization modes, as well as |
| references, see `idct`. |
| |
| Examples |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fft import dctn, idctn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idctn(dctn(y))) |
| True |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
|
|
| @_dispatch |
| def dstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, |
| workers=None, orthogonalize=None): |
| """ |
| Return multidimensional Discrete Sine Transform along the specified axes. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DST (see Notes). Default type is 2. |
| s : int or array_like of ints or None, optional |
| The shape of the result. If both `s` and `axes` (see below) are None, |
| `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is |
| ``numpy.take(x.shape, axes, axis=0)``. |
| If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``s[i] < x.shape[i]``, the ith dimension is truncated to length |
| ``s[i]``. |
| If any element of `shape` is -1, the size of the corresponding dimension |
| of `x` is used. |
| axes : int or array_like of ints or None, optional |
| Axes over which the DST is computed. If not given, the last ``len(s)`` |
| axes are used, or all axes if `s` is also not specified. |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized DST variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| y : ndarray of real |
| The transformed input array. |
| |
| See Also |
| -------- |
| idstn : Inverse multidimensional DST |
| |
| Notes |
| ----- |
| For full details of the DST types and normalization modes, as well as |
| references, see `dst`. |
| |
| Examples |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fft import dstn, idstn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idstn(dstn(y))) |
| True |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
|
|
| @_dispatch |
| def idstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, |
| workers=None, orthogonalize=None): |
| """ |
| Return multidimensional Inverse Discrete Sine Transform along the specified axes. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DST (see Notes). Default type is 2. |
| s : int or array_like of ints or None, optional |
| The shape of the result. If both `s` and `axes` (see below) are None, |
| `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is |
| ``numpy.take(x.shape, axes, axis=0)``. |
| If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``s[i] < x.shape[i]``, the ith dimension is truncated to length |
| ``s[i]``. |
| If any element of `s` is -1, the size of the corresponding dimension of |
| `x` is used. |
| axes : int or array_like of ints or None, optional |
| Axes over which the IDST is computed. If not given, the last ``len(s)`` |
| axes are used, or all axes if `s` is also not specified. |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized IDST variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| y : ndarray of real |
| The transformed input array. |
| |
| See Also |
| -------- |
| dstn : multidimensional DST |
| |
| Notes |
| ----- |
| For full details of the IDST types and normalization modes, as well as |
| references, see `idst`. |
| |
| Examples |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fft import dstn, idstn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idstn(dstn(y))) |
| True |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
|
|
| @_dispatch |
| def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, |
| orthogonalize=None): |
| r"""Return the Discrete Cosine Transform of arbitrary type sequence x. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DCT (see Notes). Default type is 2. |
| n : int, optional |
| Length of the transform. If ``n < x.shape[axis]``, `x` is |
| truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The |
| default results in ``n = x.shape[axis]``. |
| axis : int, optional |
| Axis along which the dct is computed; the default is over the |
| last axis (i.e., ``axis=-1``). |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized DCT variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| y : ndarray of real |
| The transformed input array. |
| |
| See Also |
| -------- |
| idct : Inverse DCT |
| |
| Notes |
| ----- |
| For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to |
| MATLAB ``dct(x)``. |
| |
| .. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct |
| correspondence with the direct Fourier transform. To recover |
| it you must specify ``orthogonalize=False``. |
| |
| For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same |
| overall factor in both directions. By default, the transform is also |
| orthogonalized which for types 1, 2 and 3 means the transform definition is |
| modified to give orthogonality of the DCT matrix (see below). |
| |
| For ``norm="backward"``, there is no scaling on `dct` and the `idct` is |
| scaled by ``1/N`` where ``N`` is the "logical" size of the DCT. For |
| ``norm="forward"`` the ``1/N`` normalization is applied to the forward |
| `dct` instead and the `idct` is unnormalized. |
| |
| There are, theoretically, 8 types of the DCT, only the first 4 types are |
| implemented in SciPy.'The' DCT generally refers to DCT type 2, and 'the' |
| Inverse DCT generally refers to DCT type 3. |
| |
| **Type I** |
| |
| There are several definitions of the DCT-I; we use the following |
| (for ``norm="backward"``) |
| |
| .. math:: |
| |
| y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left( |
| \frac{\pi k n}{N-1} \right) |
| |
| If ``orthogonalize=True``, ``x[0]`` and ``x[N-1]`` are multiplied by a |
| scaling factor of :math:`\sqrt{2}`, and ``y[0]`` and ``y[N-1]`` are divided |
| by :math:`\sqrt{2}`. When combined with ``norm="ortho"``, this makes the |
| corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``). |
| |
| .. note:: |
| The DCT-I is only supported for input size > 1. |
| |
| **Type II** |
| |
| There are several definitions of the DCT-II; we use the following |
| (for ``norm="backward"``) |
| |
| .. math:: |
| |
| y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right) |
| |
| If ``orthogonalize=True``, ``y[0]`` is divided by :math:`\sqrt{2}` which, |
| when combined with ``norm="ortho"``, makes the corresponding matrix of |
| coefficients orthonormal (``O @ O.T = np.eye(N)``). |
| |
| **Type III** |
| |
| There are several definitions, we use the following (for |
| ``norm="backward"``) |
| |
| .. math:: |
| |
| y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right) |
| |
| If ``orthogonalize=True``, ``x[0]`` terms are multiplied by |
| :math:`\sqrt{2}` which, when combined with ``norm="ortho"``, makes the |
| corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``). |
| |
| The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up |
| to a factor `2N`. The orthonormalized DCT-III is exactly the inverse of |
| the orthonormalized DCT-II. |
| |
| **Type IV** |
| |
| There are several definitions of the DCT-IV; we use the following |
| (for ``norm="backward"``) |
| |
| .. math:: |
| |
| y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right) |
| |
| ``orthogonalize`` has no effect here, as the DCT-IV matrix is already |
| orthogonal up to a scale factor of ``2N``. |
| |
| References |
| ---------- |
| .. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J. |
| Makhoul, `IEEE Transactions on acoustics, speech and signal |
| processing` vol. 28(1), pp. 27-34, |
| :doi:`10.1109/TASSP.1980.1163351` (1980). |
| .. [2] Wikipedia, "Discrete cosine transform", |
| https://en.wikipedia.org/wiki/Discrete_cosine_transform |
| |
| Examples |
| -------- |
| The Type 1 DCT is equivalent to the FFT (though faster) for real, |
| even-symmetrical inputs. The output is also real and even-symmetrical. |
| Half of the FFT input is used to generate half of the FFT output: |
| |
| >>> from scipy.fft import fft, dct |
| >>> import numpy as np |
| >>> fft(np.array([4., 3., 5., 10., 5., 3.])).real |
| array([ 30., -8., 6., -2., 6., -8.]) |
| >>> dct(np.array([4., 3., 5., 10.]), 1) |
| array([ 30., -8., 6., -2.]) |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
|
|
| @_dispatch |
| def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, |
| workers=None, orthogonalize=None): |
| """ |
| Return the Inverse Discrete Cosine Transform of an arbitrary type sequence. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DCT (see Notes). Default type is 2. |
| n : int, optional |
| Length of the transform. If ``n < x.shape[axis]``, `x` is |
| truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The |
| default results in ``n = x.shape[axis]``. |
| axis : int, optional |
| Axis along which the idct is computed; the default is over the |
| last axis (i.e., ``axis=-1``). |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized IDCT variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| idct : ndarray of real |
| The transformed input array. |
| |
| See Also |
| -------- |
| dct : Forward DCT |
| |
| Notes |
| ----- |
| For a single dimension array `x`, ``idct(x, norm='ortho')`` is equal to |
| MATLAB ``idct(x)``. |
| |
| .. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct |
| correspondence with the inverse direct Fourier transform. To |
| recover it you must specify ``orthogonalize=False``. |
| |
| For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same |
| overall factor in both directions. By default, the transform is also |
| orthogonalized which for types 1, 2 and 3 means the transform definition is |
| modified to give orthogonality of the IDCT matrix (see `dct` for the full |
| definitions). |
| |
| 'The' IDCT is the IDCT-II, which is the same as the normalized DCT-III. |
| |
| The IDCT is equivalent to a normal DCT except for the normalization and |
| type. DCT type 1 and 4 are their own inverse and DCTs 2 and 3 are each |
| other's inverses. |
| |
| Examples |
| -------- |
| The Type 1 DCT is equivalent to the DFT for real, even-symmetrical |
| inputs. The output is also real and even-symmetrical. Half of the IFFT |
| input is used to generate half of the IFFT output: |
| |
| >>> from scipy.fft import ifft, idct |
| >>> import numpy as np |
| >>> ifft(np.array([ 30., -8., 6., -2., 6., -8.])).real |
| array([ 4., 3., 5., 10., 5., 3.]) |
| >>> idct(np.array([ 30., -8., 6., -2.]), 1) |
| array([ 4., 3., 5., 10.]) |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
|
|
| @_dispatch |
| def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, |
| orthogonalize=None): |
| r""" |
| Return the Discrete Sine Transform of arbitrary type sequence x. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DST (see Notes). Default type is 2. |
| n : int, optional |
| Length of the transform. If ``n < x.shape[axis]``, `x` is |
| truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The |
| default results in ``n = x.shape[axis]``. |
| axis : int, optional |
| Axis along which the dst is computed; the default is over the |
| last axis (i.e., ``axis=-1``). |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized DST variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| dst : ndarray of reals |
| The transformed input array. |
| |
| See Also |
| -------- |
| idst : Inverse DST |
| |
| Notes |
| ----- |
| .. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct |
| correspondence with the direct Fourier transform. To recover |
| it you must specify ``orthogonalize=False``. |
| |
| For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same |
| overall factor in both directions. By default, the transform is also |
| orthogonalized which for types 2 and 3 means the transform definition is |
| modified to give orthogonality of the DST matrix (see below). |
| |
| For ``norm="backward"``, there is no scaling on the `dst` and the `idst` is |
| scaled by ``1/N`` where ``N`` is the "logical" size of the DST. |
| |
| There are, theoretically, 8 types of the DST for different combinations of |
| even/odd boundary conditions and boundary off sets [1]_, only the first |
| 4 types are implemented in SciPy. |
| |
| **Type I** |
| |
| There are several definitions of the DST-I; we use the following for |
| ``norm="backward"``. DST-I assumes the input is odd around :math:`n=-1` and |
| :math:`n=N`. |
| |
| .. math:: |
| |
| y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(n+1)}{N+1}\right) |
| |
| Note that the DST-I is only supported for input size > 1. |
| The (unnormalized) DST-I is its own inverse, up to a factor :math:`2(N+1)`. |
| The orthonormalized DST-I is exactly its own inverse. |
| |
| ``orthogonalize`` has no effect here, as the DST-I matrix is already |
| orthogonal up to a scale factor of ``2N``. |
| |
| **Type II** |
| |
| There are several definitions of the DST-II; we use the following for |
| ``norm="backward"``. DST-II assumes the input is odd around :math:`n=-1/2` and |
| :math:`n=N-1/2`; the output is odd around :math:`k=-1` and even around :math:`k=N-1` |
| |
| .. math:: |
| |
| y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right) |
| |
| If ``orthogonalize=True``, ``y[-1]`` is divided :math:`\sqrt{2}` which, when |
| combined with ``norm="ortho"``, makes the corresponding matrix of |
| coefficients orthonormal (``O @ O.T = np.eye(N)``). |
| |
| **Type III** |
| |
| There are several definitions of the DST-III, we use the following (for |
| ``norm="backward"``). DST-III assumes the input is odd around :math:`n=-1` and |
| even around :math:`n=N-1` |
| |
| .. math:: |
| |
| y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left( |
| \frac{\pi(2k+1)(n+1)}{2N}\right) |
| |
| If ``orthogonalize=True``, ``x[-1]`` is multiplied by :math:`\sqrt{2}` |
| which, when combined with ``norm="ortho"``, makes the corresponding matrix |
| of coefficients orthonormal (``O @ O.T = np.eye(N)``). |
| |
| The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up |
| to a factor :math:`2N`. The orthonormalized DST-III is exactly the inverse of the |
| orthonormalized DST-II. |
| |
| **Type IV** |
| |
| There are several definitions of the DST-IV, we use the following (for |
| ``norm="backward"``). DST-IV assumes the input is odd around :math:`n=-0.5` and |
| even around :math:`n=N-0.5` |
| |
| .. math:: |
| |
| y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right) |
| |
| ``orthogonalize`` has no effect here, as the DST-IV matrix is already |
| orthogonal up to a scale factor of ``2N``. |
| |
| The (unnormalized) DST-IV is its own inverse, up to a factor :math:`2N`. The |
| orthonormalized DST-IV is exactly its own inverse. |
| |
| References |
| ---------- |
| .. [1] Wikipedia, "Discrete sine transform", |
| https://en.wikipedia.org/wiki/Discrete_sine_transform |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
|
|
| @_dispatch |
| def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, |
| workers=None, orthogonalize=None): |
| """ |
| Return the Inverse Discrete Sine Transform of an arbitrary type sequence. |
| |
| Parameters |
| ---------- |
| x : array_like |
| The input array. |
| type : {1, 2, 3, 4}, optional |
| Type of the DST (see Notes). Default type is 2. |
| n : int, optional |
| Length of the transform. If ``n < x.shape[axis]``, `x` is |
| truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The |
| default results in ``n = x.shape[axis]``. |
| axis : int, optional |
| Axis along which the idst is computed; the default is over the |
| last axis (i.e., ``axis=-1``). |
| norm : {"backward", "ortho", "forward"}, optional |
| Normalization mode (see Notes). Default is "backward". |
| overwrite_x : bool, optional |
| If True, the contents of `x` can be destroyed; the default is False. |
| workers : int, optional |
| Maximum number of workers to use for parallel computation. If negative, |
| the value wraps around from ``os.cpu_count()``. |
| See :func:`~scipy.fft.fft` for more details. |
| orthogonalize : bool, optional |
| Whether to use the orthogonalized IDST variant (see Notes). |
| Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. |
| |
| .. versionadded:: 1.8.0 |
| |
| Returns |
| ------- |
| idst : ndarray of real |
| The transformed input array. |
| |
| See Also |
| -------- |
| dst : Forward DST |
| |
| Notes |
| ----- |
| .. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct |
| correspondence with the inverse direct Fourier transform. |
| |
| For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same |
| overall factor in both directions. By default, the transform is also |
| orthogonalized which for types 2 and 3 means the transform definition is |
| modified to give orthogonality of the DST matrix (see `dst` for the full |
| definitions). |
| |
| 'The' IDST is the IDST-II, which is the same as the normalized DST-III. |
| |
| The IDST is equivalent to a normal DST except for the normalization and |
| type. DST type 1 and 4 are their own inverse and DSTs 2 and 3 are each |
| other's inverses. |
| |
| """ |
| return (Dispatchable(x, np.ndarray),) |
|
|
